Computing system for solving constrained multi-objective optimization problem with complex feasible region
By combining genetic operators and differential evolution operators in the computing system, and using non-constrained optimization, constraint-diversity optimization and constraint optimization module methods, the problems of poor adaptability and insufficient diversity in the existing technology when dealing with the constraint multi-objective optimization problem in complex feasible areas are solved, and efficient solution and obtaining diversity solutions are achieved.
Patent Information
- Application Number
- CN202510147820.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-11
- Publication Date
- 2025-05-30
AI Technical Summary
When the prior art deals with the constraint multi-objective optimization problem with complex feasible areas, the adaptability is poor, it is difficult to ensure the diversity of feasible solutions, and it is difficult for algorithms to effectively explore the target space.
A computing system is proposed, combining genetic operators and differential evolution operators, and through three modules, non-constrained optimization, constraint-diversity optimization and constraint optimization, populations are screened and evaluated from different angles to ensure the diversity and constraint satisfaction of solutions.
Effectively adapt to different types of constraint multi-objective optimization problems, ensure that the obtained feasible solutions have sufficient diversity and optimization effects, and improve the solution efficiency and accuracy of the algorithm.
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Figure CN120068629A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of constrained multi-objective optimization, and particularly to a computing system for solving constrained multi-objective optimization problems with complex feasible regions. Background Art
[0002] Multi-objective optimization problems are a common type of optimization problems in the fields of scientific research and engineering applications. This type of problem needs to take into account multiple optimization objectives simultaneously and requires the algorithm to give a set of alternative solutions that can make different trade-offs among multiple optimization objectives. For example, in the design of unmanned aerial vehicles, we not only need to consider the weight of the airframe, but also need to consider multiple objectives such as its endurance time and manufacturing cost, and need to make decisions based on the preference degree for different objectives according to specific requirements. When the specific requirements are not yet determined, we require the algorithm to provide a set of compromise solutions that can cover potential requirements, so as to help make a quick decision after the specific requirements are determined.
[0003] Constrained multi-objective optimization problems are a common and special type of multi-objective optimization problems. These problems have some constraint conditions that need to be satisfied, and only the solutions that satisfy all the constraint conditions are feasible solutions, otherwise they are infeasible solutions. For example, in the design of unmanned aerial vehicles, while considering optimization objectives such as the weight of the airframe, endurance time, and manufacturing cost, we must also consider whether the strength of the components meets the requirements. If the requirements cannot be met, then even if the other objectives perform excellently, they are not advisable. Compared with general multi-objective optimization problems, the optimization difficulty of constrained multi-objective optimization problems increases significantly. Due to the limitations brought by the constraint conditions, it often leads to the algorithm being difficult to fully explore the objective space and thus find the Pareto optimal solution that satisfies the constraint conditions.
[0004] Currently, for this problem, there is a solution method based on a dual optimization strategy. This solution method is to simultaneously execute a constraint-based optimization strategy and an unconstrained-based optimization strategy during an iterative process, to get rid of the limitation of the constraint on the search scope through the unconstrained-based optimization strategy, and to ensure that the obtained solutions can satisfy the constraint conditions through the constraint-based optimization strategy, so as to find the Pareto optimal solution that satisfies the constraint conditions while minimizing the impact of the constraints as much as possible.
[0005] The main problems of this method are as follows:
[0006] 1. Poor adaptability to different types of constrained multi-objective optimization problems. For many problems, their constraint conditions often present characteristics such as diversification and unpredictability, which will lead to the complexity of the distribution of the feasible regions in the objective space, such as being discrete, narrow, tiny, etc., resulting in the algorithm being difficult to find these feasible regions and difficult to achieve the solution.
[0007] 2. It is difficult to ensure the diversity of the obtained feasible solutions. In the existing algorithms, since the constraint-based optimization strategy only considers diversity when a sufficient number of feasible solutions are collected, this method cannot avoid the limitation of the search scope by the constraint conditions, resulting in the difficulty for the algorithm to find feasible solutions with sufficient diversity.
[0008] In summary, the present application proposes a computing system for solving constrained multi-objective optimization problems with complex feasible regions. Summary of the Invention
[0009] The object of the present invention is to address the problem of poor adaptability of the dual optimization strategy to different types of constrained multi-objective optimization problems in the background art, and to propose a computing system for solving constrained multi-objective optimization problems with complex feasible regions.
[0010] The technical solution of the present invention: A computing system for solving constrained multi-objective optimization problems with complex feasible regions, comprising:
[0011] An initialization module that generates an initial population that meets the requirements of the problem to be solved according to the input parameters;
[0012] A child generation module that generates a child population using an evolutionary strategy based on genetic operators and differential evolution operators on the basis of the initial population or the population returned after one iteration;
[0013] An unconstrained optimization module that, without considering the constraint conditions, evaluates the convergence and diversity of each individual in the objective space according to the values obtained by each individual on each objective, performs screening according to the evaluation results, and stores the individuals obtained after screening;
[0014] A constraint-diversity optimization module that evaluates the diversity of each individual in the objective space according to the values obtained by each individual on each objective, and performs screening on the population considering the satisfaction of each individual with the constraint conditions on the basis of meeting the diversity requirements, and stores the individuals obtained after screening;
[0015] A constraint optimization module that screens the population according to the satisfaction of each individual with the constraint conditions, and after screening, evaluates the convergence and diversity of each individual in the objective space according to the values obtained by each individual on each objective, performs re-screening according to the evaluation results, and stores the individuals obtained after screening;
[0016] A result output module that outputs the variable data and objective value data of the population stored in the constraint optimization module after the optimization is completed, and displays its optimization effect in an intuitive manner.
[0017] Optionally, the computing system stores elite individuals using three external archives, each with a different screening strategy, so as to maximize the retention of evolutionary potential, enhance the robustness of the algorithm, and effectively solve various types of constrained multi-objective optimization problems with complex feasible regions.
[0018] Optionally, the offspring generation module simultaneously uses genetic operators and differential evolution operators to generate offspring with a 50% probability for each, thereby increasing the diversity of the generated offspring.
[0019] The offspring generation module independently generates the offspring populations from the non-constrained optimization module, the constraint-diversity optimization module, and the parent population saved in the optimization module. After generation, the sub-populations are incorporated into the same population pool to form a new generation of population;
[0020] Optionally, the non-constrained optimization module first screens the new generation of population and the previous generation of population saved in this module according to the Pareto dominance method to obtain a new population, and then uses the environmental selection method to further screen the obtained new population and save it into the non-constrained optimization module.
[0021] Optionally, in the constraint-diversity optimization module, first screen the new generation of population and the previous generation of population saved in this module according to the constraint-Pareto dominance method to obtain a new population, and then use the constraint-diversity environmental selection method to further screen the obtained new population and save it into the constraint-diversity optimization module.
[0022] Optionally, based on the constraint-Pareto dominance method, a comparison of feasibility is added to determine the dominance relationship, and its calculation formula is:
[0023] Let individual a and individual b be two different individuals in the population. When and only when
[0024]
[0025] is satisfied, it is said that individual a constraint-Pareto dominates individual b, where M is the number of objectives, f i (a represents the value obtained by individual a on the i-th objective, f i (b) represents the value obtained by individual b on the i-th objective, CV(a) represents the feasibility value of individual a, and CV(b) represents the feasibility value of individual b.
[0026] Optionally, the constraint-diversity environmental selection method uses the method of establishing reference vectors in the objective space to divide the objective space into several sub-spaces, and each sub-space conducts independent searches and only retains one individual with the optimal feasibility value;
[0027] The fitness of individuals in each subspace is calculated through a fitness calculation formula, and the individuals to be retained are determined based on the fitness. The calculation formula is as follows:
[0028]
[0029] where fitness ij represents the fitness value of the i-th individual in the subspace formed by the j-th reference vector, h represents the minimum angular distance between reference vectors, CV(x i ) represents the feasibility value of the i-th individual, and d ij represents the angular distance between the i-th individual and the j-th reference vector.
[0030] Optionally, in the constraint optimization module, first, according to the constraint-Pareto dominance method, the new generation population and the previous generation population saved by this module are screened to obtain a new population. Then, the feasibility selection method is used to further screen the obtained new population. Finally, through the environmental selection method, the final population is obtained and saved into the constraint-diversity module.
[0031] Optionally, the unconstrained optimization module uses the environmental selection method of NSGA-III to screen individuals, and the constraint optimization module uses the environmental selection method of SPEA2 to screen individuals.
[0032] Optionally, in the result output module, an image is generated to visually display the values of the obtained individuals on each target, providing assistance for users to make decisions.
[0033] Compared with the prior art, the present application includes at least one of the following beneficial technical effects:
[0034] The initialization module can quickly generate an initial population that meets the requirements. The offspring generation module combines genetic operators and differential evolution operators to generate offspring with a 50% probability each, which can effectively explore the solution space, enhance population diversity and evolutionary ability. At the same time, the offspring populations are independently generated by the parent populations saved by different modules and incorporated into the same pool, increasing the richness of the population and facilitating the search for better solutions.
[0035] Multi-dimensional optimization strategy: Through three modules of unconstrained optimization, constraint-diversity optimization, and constraint optimization, the population is screened and evaluated from different perspectives. The unconstrained optimization module focuses on the convergence and diversity of the objective space, providing a basis for subsequent optimization; the constraint-diversity optimization module takes into account both diversity and constraint conditions while considering diversity, ensuring that the optimization direction meets both diversity requirements and does not violate constraints; the constraint optimization module first screens according to constraint conditions and then evaluates convergence and diversity, ensuring that the final solution not only meets the constraints but also has good performance in the objective space.
[0036] Effective elite individual storage: By using three external archives with different screening strategies, it helps to retain different types of excellent individuals, avoid losing valuable solutions during the evolution process, and enhance the global search ability of the algorithm.
[0037] Reasonable screening methods: Different screening methods are adopted for each module. For example, the unconstrained optimization module uses the Pareto dominance method and the environmental selection method of NSGA-III, the constraint-diversity optimization module uses the constraint-Pareto dominance method and the constraint-diversity environmental selection method, and the constrained optimization module uses the constraint-Pareto dominance method, the feasibility selection method, and the environmental selection method of SPEA2. These methods are highly targeted and can effectively screen according to the characteristics of the problem at different stages, improving the solution efficiency and accuracy of the algorithm.
[0038] Intuitive result display: The result output module visually displays the values of individuals on each objective by generating images, providing clear decision-making basis for users, reducing the decision-making difficulty, and improving the decision-making efficiency.
[0039] The computing system proposed by the present invention can effectively adapt to different types of constrained multi-objective optimization problems. Even in the face of complex and diverse constraint conditions and complexly distributed feasible regions, it can successfully solve the problems. At the same time, through unique optimization strategies, it avoids excessive restrictions of constraints on the search range, ensures that the obtained feasible solutions have sufficient diversity, and provides better solutions for practical applications. Brief Description of the Drawings
[0040] Figure 1 It is a principle block diagram of a computing system for solving constrained multi-objective optimization problems with complex feasible regions;
[0041] Figure 2 It is a data display diagram of the final optimization objective values when the number of objectives is 2;
[0042] Figure 3 It is a data display diagram of the final optimization objective values when the number of objectives is 3. Detailed Embodiments
[0043] The technical solutions of the present invention will be further described below in conjunction with the drawings and specific embodiments.
[0044] Embodiment
[0045] Such as Figure 1As shown, a computing system proposed by the present invention for solving constrained multi-objective optimization problems with complex feasible regions stores elite individuals using three external archives. Each external archive adopts a different screening strategy, so as to maximize the retention of evolutionary potential, enhance the robustness of the algorithm, and effectively solve various types of constrained multi-objective optimization problems with complex feasible regions. The computing system includes an initialization module, a child generation module, an unconstrained optimization module, a constraint-diversity optimization module, a constraint optimization module, and a result output module. Each module will be described in detail below.
[0046] In this embodiment, the initialization module generates an initial population that meets the requirements of the problem to be solved according to the input parameters. The input requirements of the system are: population size, number of variables to be optimized, number of objectives, problem function, number of iterations. After the input is completed, the system will first enter the initialization module. In the initialization module, the system will perform the following operations in sequence:
[0047] 1. According to the population size and the number of variables to be optimized of the input problem function, randomly generate an initial population, with its upper limit uniformly set to 1 and its lower limit uniformly set to zero;
[0048] 2. Initialize the unconstrained optimization module, the constraint-diversity optimization module, and the constraint optimization module according to the population size of the input problem function, and set their capacities to N, where N is the population size, and randomly fill the initial population into the three optimization modules;
[0049] After the initialization is completed, it enters the child generation module. The child generation module generates a child population using an evolutionary strategy based on genetic operators and differential evolution operators on the basis of the initial population or the population returned after one iteration. The child generation module simultaneously uses genetic operators and differential evolution operators to generate children with a probability of 50% each, so as to improve the diversity of the generated children.
[0050] Among them, the child population of the child generation module is independently generated by the unconstrained optimization module, the constraint-diversity optimization module, and the parent population stored in the optimization module respectively. After the generated sub-populations are incorporated into the same population pool to form a new generation population;
[0051] In the child generation module, the system will perform the following operations in sequence:
[0052] 1. Randomly select two individuals from the unconstrained optimization module;
[0053] 2. Randomly use genetic operators or differential evolution operators to generate the offspring of these two individuals;
[0054] 3. Repeat steps 1 and 2 until all individuals in the unconstrained optimization module are selected;
[0055] 4. Randomly select two individuals in the constraint-diversity optimization module;
[0056] 5. Randomly use genetic operators or differential evolution operators to generate offspring of these two individuals;
[0057] 6. Repeat steps 4 and 5 until all individuals in the constraint-diversity optimization module are selected;
[0058] 7. Randomly select two individuals in the constraint optimization module;
[0059] 8. Randomly use genetic operators or differential evolution operators to generate offspring of these two individuals;
[0060] 9. Repeat steps 7 and 8 until all individuals in the constraint optimization module are selected;
[0061] 10. Merge all the generated offspring populations to form a new generation population.
[0062] Without considering the constraint conditions, the unconstrained optimization module evaluates the convergence and diversity of each individual in the objective space according to the values obtained by each individual on various objectives, screens according to the evaluation results, and stores the individuals obtained after screening; for the unconstrained optimization module, first, according to the Pareto domination method, screen the new generation population and the previous generation population saved by this module to obtain a new population, and then use the environmental selection method to further screen the obtained new population and save it into the unconstrained optimization module; when entering the unconstrained optimization module, the system will perform the following operations in sequence:
[0063] 1. Mix the new generation population with the previous generation population stored in the unconstrained optimization module, and use the non-dominated sorting method to calculate the domination relationship between each other by pairwise comparison of the objective values of each individual in the population;
[0064] 2. Delete the dominated individuals from the population
[0065] 3. Count the number of remaining individuals. If the remaining quantity is less than the capacity of this module, randomly copy the remaining individuals until their quantity is equal to the capacity of this module;
[0066] 4. Use the environmental selection method based on NSGA-II to screen individuals
[0067] 5. Calculate the maximum value point of the objective space according to the objective values of each individual in the population;
[0068] 6. Establish a normalized space according to the maximum value point and the minimum value point of the objective space;
[0069] 7. Normalize the objective values of all individuals in the population;
[0070] 8. Calculate the Euclidean distance between every two individuals in the population as their crowding distance;
[0071] 9. Delete the individual with the smallest crowding distance in the population until the remaining population size is equal to the capacity of the unconstrained module;
[0072] 10. Store and output the remaining population;
[0073] In this embodiment, the constraint-diversity optimization module evaluates the diversity of each individual in the objective space according to the values obtained by each individual on each objective, and screens the population considering the satisfaction of each individual with the constraint conditions on the basis of meeting the diversity requirements, and stores the individuals obtained after screening; in the constraint-diversity optimization module, first, according to the constraint-Pareto dominance method, screen the new generation population and the previous generation population saved by the module to obtain a new population, and then adopt the constraint-diversity environmental selection method to further screen the obtained new population and save it into the constraint-diversity optimization module.
[0074] Among them, on the basis of the constraint-Pareto dominance method, a comparison of feasibility is added to determine the dominance relationship, and its calculation formula is:
[0075] Let individual a and individual b be two different individuals in the population. When and only when
[0076]
[0077] is satisfied, it is said that individual a constraint-Pareto dominates individual b, where M is the number of objectives, f i (a represents the value obtained by individual a on the i-th objective, f i (b) represents the value obtained by individual b on the i-th objective, CV(a) represents the feasibility value of individual a, and CV(b) represents the feasibility value of individual b.
[0078] In addition, the constraint-diversity environmental selection method adopts the method of establishing a reference vector in the objective space to divide the objective space into several subspaces, and each subspace conducts independent search and only retains one individual with the optimal feasibility value;
[0079] Calculate the fitness of the individuals in each subspace through a fitness calculation formula, and determine the retained individuals according to the fitness. Its calculation formula is:
[0080]
[0081] where fitness ijdenotes the fitness value of the \(i\)-th individual in the subspace formed by the \(j\)-th reference vector, \(h\) represents the minimum angular distance between reference vectors, \(CV(x i ) represents the feasibility value of the \(i\)-th individual, \(d ij represents the angular distance between the \(i\)-th individual and the \(j\)-th reference vector. Entering the constraint-diversity optimization module, the system will perform the following operations in sequence:
[0082] 1. Mix the new generation population with the previous generation population stored in the constraint-diversity optimization module, and use the constraint Pareto dominance method to calculate the dominance relationship between each other by pairwise comparison of the objective values of the individuals in the population;
[0083] 2. Delete the dominated individuals from the population
[0084] 3. Calculate the maximum value point of the objective space according to the objective values of each individual in the population;
[0085] 4. Normalize the objective values of all individuals in the population;
[0086] 5. Establish reference vectors in the normalized space;
[0087] 6. Calculate the minimum angular distance between all reference vectors;
[0088] 7. Select a reference vector, count all individuals whose angular distance from it is less than the minimum angular distance between reference vectors, and put them into a temporary archive. If there are no individuals within this range, put the individual with the smallest angular distance from this reference vector into the temporary archive;
[0089] 8. Select the individual with the lowest degree of constraint violation in the temporary archive and save it into the constraint-diversity optimization module. If there are multiple individuals that do not violate any constraints, select the individual with the smallest angular distance from the reference vector among these individuals that do not violate any constraints and save it into the constraint-diversity optimization module;
[0090] 9. Repeat steps 7 and 8 until all reference vectors are selected;
[0091] 10. Store and output the saved population.
[0092] In this application, the constraint optimization module screens the population according to the satisfaction of each individual with respect to the constraint conditions, and after screening, evaluates the convergence and diversity of each individual in the objective space based on the values obtained on each objective, performs another screening according to the evaluation results, and stores the individuals obtained after screening; in the constraint optimization module, first, according to the constraint-Pareto dominance method, the new generation population and the previous generation population saved by this module are screened to obtain a new population, then the feasibility selection method is used to further screen the obtained new population, and finally, through the environmental selection method, the final population is obtained and saved into the constraint-diversity module.
[0093] It should be noted that the non-constraint optimization module uses the environmental selection method of NSGA-III to screen individuals, and the constraint optimization module uses the environmental selection method of SPEA2 to screen individuals.
[0094] When entering the constraint optimization module, the system will perform the following operations in sequence:
[0095] 1. Mix the new generation population with the previous generation population stored in the constraint optimization module, and use the constraint Pareto dominance method to calculate the dominance relationship between each other by pairwise comparison of the objective values of each individual in the population;
[0096] 2. Delete the dominated individuals from the population
[0097] 3. Count the number of remaining individuals. If the remaining quantity is less than the capacity of this module, randomly copy the remaining individuals until their quantity is equal to the capacity of this module;
[0098] 4. Count the constraint violation situations of the remaining individuals and sort them in ascending order according to the degree of constraint violation;
[0099] 5. Delete all individuals with serial numbers higher than the capacity of the constraint optimization module and having constraint violation situations;
[0100] 6. Use the environmental selection method based on SPEA2 to screen individuals
[0101] 7. Calculate the maximum value point of the objective space according to the objective values of each individual in the population;
[0102] 8. Establish a normalized space based on the maximum value point and the minimum value point of the objective space;
[0103] 9. Normalize the objective values of all individuals in the population;
[0104] 10. Calculate the Euclidean distance between each pair of individuals in the population as their crowding distance;
[0105] 11. Delete the individual with the smallest crowding distance in the population
[0106] 12. Recalculate the crowding distance between each individual;
[0107] 13. Repeat steps 11 and 12 until the number of remaining individuals is equal to the capacity of the non-constrained module
[0108] 14. Store and output the remaining population.
[0109] The result output module outputs the variable data and objective value data of the population stored in the constrained optimization module after optimization, and displays its optimization effect in an intuitive way. The generated image intuitively shows the values of the obtained individuals on each objective, providing help for the user to make decisions. The system will determine whether the current iteration number reaches the maximum iteration number. If not, the system will return to the offspring generation module, use the current population as the parent population for generating offspring, and perform a new round of loop until the current iteration number reaches the maximum iteration number. After reaching the maximum iteration number, the system enters the result output module and will perform the following operations in sequence:
[0110] 1. Plot the objective value data of the output population;
[0111] 2. Save the variable data of the final output population in the dec.mat file;
[0112] 3. Save the objective value data of the final output population in the obj.mat file;
[0113] 4. Exit the system program.
[0114] Experimental case
[0115] To run this system, first start Matlab, enter the directory where the system is located, then enter CMOEACS_main() in the command window, and enter a series of parameters and problem functions in the parentheses. The inputs are as follows in sequence:
[0116] 1. Population size;
[0117] 2. Number of optimization objectives;
[0118] 3. Number of variables to be optimized;
[0119] 4. Problem function;
[0120] 5. Maximum number of iterations;
[0121] After the input is completed, press the enter key. At this time, the system will start running, and whenever 10% of the running progress is completed, a prompt "Current progress XX%" will pop up, where 'XX%' represents the percentage of the current progress completed (such as 10%, 20%, 30%).
[0122] When the system operation ends, a prompt "Current progress 100%" will pop up, and the data of the variables to be optimized in the final output population will be saved in the dec.mat file, and the data of the objective values in the final output population will be saved in the obj.mat file. The save path of this file is the same as the directory where the system is located. It can be double-clicked to load it into the Matlab workspace and then opened from the workspace for viewing.
[0123] When the system operation ends, a frame will pop up, showing the data of the final obtained population on the objective values and the corresponding values of the variables to be optimized. When the number of objectives is 2, the data of the final optimized objective values is displayed in a two-dimensional graph as attached Figure 2 , where the two number axes respectively represent the values obtained on the two objectives; when the number of objectives is 3, the data of the final optimized objective values is displayed in a three-dimensional graph as attached Figure 3 , where the three number axes respectively represent the values obtained on the three objectives.
[0124] The above specific embodiments are only several alternative embodiments of the present invention. Based on the technical solution of the present invention and the relevant revelations of the above embodiments, those skilled in the art can make various alternative improvements and combinations to the above specific embodiments.
Claims
1. A computing system for solving constrained multi-objective optimization problems with complex feasible regions, characterized in that: include: Initialization module, which generates an initialization population that meets the requirements of solving the problem according to the input parameters; The offspring generation module generates offspring populations based on the initial population or the population returned after one iteration by adopting an evolutionary strategy based on genetic operators and differential evolution operators; The unconstrained optimization module evaluates the convergence and diversity of each individual in the target space based on the values obtained by each individual on each target without considering the constraints, performs screening based on the evaluation results, and stores the individuals obtained after screening; The constraint-diversity optimization module evaluates the diversity of each individual in the target space according to the value obtained by each individual on each target, and screens the population based on the satisfaction of the constraints of each individual on the basis of meeting the diversity requirements, and stores the individuals obtained after screening; The constrained optimization module screens the population according to whether each individual satisfies the constraint conditions, and after screening, evaluates the convergence and diversity of each individual in the target space according to the values obtained by each individual on each target, screens again according to the evaluation results, and stores the individuals obtained after screening; The result output module outputs the variable data and target value data of the population stored in the constraint optimization module after the optimization is completed, and displays its optimization effect in an intuitive way.
2. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: The computing system uses three external archives to store elite individuals, and each external archive uses a different screening strategy.
3. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: The offspring generation module uses both genetic operators and differential evolution operators to generate offspring with a probability of 50% each; The offspring generation module has its offspring populations independently generated by the unconstrained optimization module, the constrained-diversity optimization module, and the parent populations saved in the optimization module. The generated sub-populations are incorporated into the same population pool to form a new generation of populations.
4. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: The unconstrained optimization module first screens the new generation population and the previous generation population saved by the module according to the Pareto dominance method to obtain a new population, and then uses the environmental selection method to further screen the obtained new population and save it in the unconstrained optimization module.
5. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: In the constraint-diversity optimization module, the new generation population and the previous generation population saved by the module are first screened according to the constraint-Pareto dominance method to obtain a new population, and then the constraint-diversity environment selection method is used to further screen the obtained new population and save it in the constraint-diversity optimization module.
6. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: On the basis of the constraint-Pareto dominance method, a comparison of feasibility is added to determine the dominance relationship, and the calculation formula is: Assume that individuals a and b are two different individuals in the population if and only if When satisfied, individual a is said to constrain -Pareto dominate individual b, where M is the number of targets, f i (a) represents the value obtained by individual a on the i-th target, f i (b) represents the value obtained by individual b on the i-th target, CV(a) represents the feasibility value of individual a, and CV(b) represents the feasibility value of individual b.
7. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: The constraint-diversity environment selection method adopts the method of establishing a reference vector in the target space, dividing the target space into several subspaces, each subspace independently conducts a search and only retains an individual with the best feasibility value; The fitness of individuals in each subspace is calculated through a fitness calculation formula, and the retained individuals are determined based on the fitness. The calculation formula is: Among them fitness ij represents the fitness value of the i-th individual in the subspace formed by the j-th reference vector, h represents the minimum angular distance between the reference vectors, CV(x i ) represents the feasibility value of the i-th individual, d ij represents the angular distance between the i-th individual and the j-th reference vector.
8. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: In the constraint optimization module, the new generation population and the previous generation population saved by the module are first screened according to the constraint-Pareto dominance method to obtain a new population, and then the feasibility selection method is used to further screen the obtained new population. Finally, the final population is obtained through the environmental selection method and saved in the constraint-diversity module.
9. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: The unconstrained optimization module uses the environmental selection method of NSGA-III to screen individuals, and the constrained optimization module uses the environmental selection method of SPEA2 to screen individuals.
10. A computing system for solving constrained multi-objective optimization problems with complex feasible regions according to claim 1, characterized in that: In the result output module, images are generated to display the obtained individual values on various targets in an intuitive manner, thereby helping users make decisions.